
As shown in [1]}, \(\widetilde{\mathbf {N}}_1\)  and \(\widetilde{\mathbf {N}}_0\) 
satisfy the following properties:
\(0 \prec \widetilde{\mathbf {N}}_1 &= \left(\mathbf {N}_1^{-1}+\mu _1^{-1}\mathbf {M}_1\right)^{-1}\preceq \mathbf {N}_1, \) 
\(\widetilde{\mathbf {N}}_1 &\preceq \widetilde{\mathbf {N}}_0 \preceq \mathbf {N}_0, \) 
\(\left|\frac{\mathbf {B}^*+\widetilde{\mathbf {N}}_1}{\widetilde{\mathbf {N}}_1}\right|&=\left|\frac{\mathbf {B}^*+\mathbf {N}_1}{\mathbf {N}_1}\right| \) 
